To determine the range of acceptable lengths for the metal rod given that its ideal length is [tex]$20.5 \, \text{cm}$[/tex] and it may vary by at most [tex]$0.045 \, \text{cm}$[/tex], follow these steps:
1. Determine the maximum allowed deviation:
The measured length can vary from the ideal length by [tex]$0.045 \, \text{cm}$[/tex].
2. Calculate the lower limit of the range:
Subtract the maximum deviation from the ideal length:
[tex]\[
20.5 \, \text{cm} - 0.045 \, \text{cm} = 20.455 \, \text{cm}
\][/tex]
3. Calculate the upper limit of the range:
Add the maximum deviation to the ideal length:
[tex]\[
20.5 \, \text{cm} + 0.045 \, \text{cm} = 20.545 \, \text{cm}
\][/tex]
4. Express the range of acceptable lengths:
The acceptable length of the rod should be at least [tex]$20.455 \, \text{cm}$[/tex] and at most [tex]$20.545 \, \text{cm}$[/tex].
Therefore, the range of acceptable lengths for the rod is:
[tex]\[
20.455 \leq x \leq 20.545
\][/tex]
By comparing this result to the given choices, the correct answer is:
[tex]\[
\boxed{D. \, 20.455 \leq x \leq 20.545}
\][/tex]
